Basic invariants
| Dimension: | $5$ |
| Group: | $S_5$ |
| Conductor: | \(38348997241\)\(\medspace = 113^{2} \cdot 1733^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 5.5.195829.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_5$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_5$ |
| Projective stem field: | Galois closure of 5.5.195829.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 a + 21 + \left(2 a + 8\right)\cdot 37 + \left(3 a + 11\right)\cdot 37^{2} + \left(27 a + 23\right)\cdot 37^{3} + \left(35 a + 27\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 9 a + 22 + \left(34 a + 28\right)\cdot 37 + \left(33 a + 20\right)\cdot 37^{2} + \left(9 a + 17\right)\cdot 37^{3} + \left(a + 32\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 a + 27 + \left(27 a + 21\right)\cdot 37 + \left(34 a + 25\right)\cdot 37^{2} + \left(16 a + 4\right)\cdot 37^{3} + \left(7 a + 9\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 27\cdot 37^{2} + 27\cdot 37^{3} + 19\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 a + 10 + \left(9 a + 14\right)\cdot 37 + \left(2 a + 26\right)\cdot 37^{2} + 20 a\cdot 37^{3} + \left(29 a + 22\right)\cdot 37^{4} +O(37^{5})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 5 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 5 }$ | Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.